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2010•Ubaya Repository (University of Surabaya)Open access

The Metric Dimension of Amalgamation of Cycles

Hazrul Iswadi, Edy Tri Baskoro, A.N.M. Salman, Rinovia Simanjuntak, Jalan Ganesha, Jalan Raya

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Abstract

For an ordered set {}kwwwW...,,, 21 = of vertices and a vertex v in a connected graph G, the representation of v with respect to W is the ordered k-tuple ( ) ( ) ( ) () (),,...,,,,, 21 kwvdwvdwvdWvr = | where ()yxd, represents the distance between the vertices x and y. The set W is called a resolving set for G if every vertex of G has a distinct representation. A resolving set containing a minimum number of vertices is called a basis for G. The dimension of G, denoted by (),dim G is the number of vertices ISWADI, BASKORO, SALMAN and SIMANJUNTAK 20 in a basis of G. Let {}iG be a finite collection of graphs and each iG has a fixed vertex oiv called a terminal. The amalgamation Amal {}oii vG, is formed by taking all of the iG ’s and identifying their terminals. In this paper, we determine the metric dimension of amalgamation of cycles. 1.

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For an ordered set {}kwwwW...,,, 21 = of vertices and a vertex v in a connected graph G, the representation of v with respect to W is the ordered k-tuple ( ) ( ) ( ) () (),,...,,,,, 21 kwvdwvdwvdWvr = | where ()yxd, represents the distance between the vertices x and y. The set W is called a resolving set for G if every vertex of G has a distinct representation. A resolving set containing a minimum number of vertices is called a basis for G. The dimension of G, denoted by (),dim G is the number of vertices ISWADI, BASKORO, SALMAN and SIMANJUNTAK 20 in a basis of G. Let {}iG be a finite collection of graphs and each iG has a fixed vertex oiv called a terminal. The amalgamation Amal {}oii vG, is formed by taking all of the iG ’s and identifying their terminals. In this paper, we determine the metric dimension of amalgamation of cycles. 1.

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Available abstract

For an ordered set {}kwwwW...,,, 21 = of vertices and a vertex v in a connected graph G, the representation of v with respect to W is the ordered k-tuple ( ) ( ) ( ) () (),,...,,,,, 21 kwvdwvdwvdWvr = | where ()yxd, represents the distance between the vertices x and y. The set W is called a resolving set for G if every vertex of G has a distinct representation. A resolving set containing a minimum number of vertices is called a basis for G. The dimension of G, denoted by (),dim G is the number of vertices ISWADI, BASKORO, SALMAN and SIMANJUNTAK 20 in a basis of G. Let {}iG be a finite collection of graphs and each iG has a fixed vertex oiv called a terminal. The amalgamation Amal {}oii vG, is formed by taking all of the iG ’s and identifying their terminals. In this paper, we determine the metric dimension of amalgamation of cycles. 1.

Key concepts: Combinatorics, Vertex (graph theory), Metric dimension, Mathematics, Dimension (graph theory), Graph, Bound graph, Discrete mathematics

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